Optimal learning rates for Kernel Conjugate Gradient regression

TL;DR

This paper establishes optimal convergence rates for kernel conjugate gradient regression with early stopping, matching theoretical lower bounds and extending results to semi-supervised settings.

Contribution

It provides the first matching upper bounds for kernel conjugate gradient regression rates, linking regularity and data dimensionality, and extends to semi-supervised learning.

Findings

Rates depend on target function regularity and data intrinsic dimension.

Upper bounds match known lower bounds up to a log factor.

Results align with state-of-the-art for SVMs and linear regularization.

Abstract

We prove rates of convergence in the statistical sense for kernel-based least squares regression using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is directly related to Kernel Partial Least Squares, a regression method that combines supervised dimensionality reduction with least squares projection. The rates depend on two key quantities: first, on the regularity of the target regression function and second, on the intrinsic dimensionality of the data mapped into the kernel space. Lower bounds on attainable rates depending on these two quantities were established in earlier literature, and we obtain upper bounds for the considered method that match these lower bounds (up to a log factor) if the true regression function belongs to the reproducing kernel Hilbert space. If this assumption is not fulfilled, we obtain similar convergence rates provided additional unlabeled data are available. The order of the learning rates match state-of-the-art results that were recently obtained for least squares support vector machines and for linear regularization operators.